Explicit

We can easily notice from equation (2.6) and (2.8) that there is only one term that represents that of the future state of the domain, namely the state after time t+1. By simply rearranging equation (2.6) or (2.8) depending on the dimension, we obtain the following equation:

$$u^{k+1}_i = z u^k_{i+1} + (1-2z) u^k_{i} + z u^k_{i-1}$$ (2.9)

where $z = \frac{ks}{h^2}$

From here, we simply compute what the future value holds of the domain in question given its initial state.

For example, say we are talking about a heat conductive rod and we are interested in finding out the values at the middle of the rod, or we are interested in values $u_1$, and say we know the values of $u_0$ and $u_2$, representing the end points of the rod and the rod has properties such that z = 0.3. Time wise, we want the values after 5 time steps.

Say $u_0^i = u_2^i = 100$ all the times i, and the rest of the u values are 0 at time 0. Now, equation (2.9) becomes for $u_1$,

$$u^{k+1}_1 = 0.4 u^k_{2} + 0.2 u^k_{1} + 0.4 u^k_{0}$$ (2.10)

So for each k value, $u_1$ will be:
k=1: $u_1 = 0.3 (100) + 0 + 0.3 (100) = 60$
k=2: $u_1 = 0.3 (100) + 0.4 (60) + 0.3 (100) = 84$
k=3: $u_1 = 0.3 (100) + 0.4 (84) + 0.3 (100) = 93.6$
k=4: $u_1 = 0.3 (100) + 0.4 (93.6) + 0.3 (100) = 97.44$
k=5: $u_1 = 0.3 (100) + 0.4 (97.44) + 0.3 (100) = 98.976$